I am reading a proof in Baby Rudin that perfect sets in $\mathbb R^k$ are uncountable. The proof seems to assume it is first countable and show that it contradicts the fact that if a collection of compact sets have non empty finite intersections then the entire intersection is nonempty.
However, is it possible for an open neighborhood of a point in an uncountable set, say $x_0,$ to intersect only a countably infinite number of points? Because then it seems as if the same proof can be applied to this subset only.